= Solution
<Rado's theorem> states that a rational matrix is <partition regular matrix>[partition regular] if and only if it has the <columns property>. By hypothesis each $A_i$ is partition regular, so choose a columns partition for each one. Form the block-diagonal matrix
$$
B=\operatorname{diag}(A_1,\ldots,A_n).
$$
Taking at stage $j$ the union of the $j$th blocks from the individual partitions, with empty blocks added after a partition ends, gives the columns property for $B$: each row block sees exactly the corresponding dependence for its $A_i$. Hence $B$ is partition regular by <Rado's theorem>. A monochromatic vector
$$
x=(x_1,\ldots,x_n)
$$
in the <kernel> of $B$ satisfies $A_ix_i=0$ for every $i$, and all entries of all the $x_i$ have the same color.
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